190 XIII INTEGRATION 13.9, Problems 13 and 24). Show that A(A) is either 0 or 1. (Suppose that d = A(A) e ]0,1 [. Show that, for each component interval J of I — En, we have A(A n J) ^ (2dl(l 4- <0)A(J). For this purpose use (a) above and Problem l(c) of Section 13.21 to show that dt r1 , v dt rd dt for suitably chosen integers /?, q, r, s1 with /w — qr = ± 1. Use Problem 2 to complete the proof.) (c) Show that the measure p. — g • A, where g(x) = !/(!+ x), is invariant with respect to p. Deduce from (b) and the ergodic theorem (Section 13.9, Problems 13 and 24) that, for every A-integrable function / on I and almost all irrational x e I, we have n~*oo n log 2 JQ 1 -f t (Gauss-Kuzmin formula). (d) Deduce from (c) that, for each integer p ^ 1, if vn(x, p) is the number of indices k<^n such that qk(x) —p(x irrational), then log lOg 2 for almost all irrational x e I. (e) Deduce from (c) that oo / 1 \ log n/log 2 lim fofrte&c) • • -^W)1'" = FT 11 + -TTV n-»« n = i \ n2 + 2n] for almost all irrational x. 15. THE LEBESGUE-NIKODYM THEOREM AND THE ORDER RELATION ON MR(X) (13.15.1) Let \JL and v be two positive measures on a locally compact space X, such that v ^ ju. Then there exists a locally n-integrable function g such that We distinguish two cases. (I) Suppose first that v is bounded. Since the function 1 is then v-integ- rable, it follows from (1 3.1 1 .2.2) that for every /e Jf R(X) (13.15.1.1) Deduce from (a) that if r=*p/q and r' — p'lq' are two rational numbers belonging